Summary

This file introduces Uniform-Iteration Algebras

• Definition 12 Uniform-Iteration Algebras

Code

module UniformIterationAlgebra {o ℓ e} (D : ExtensiveDistributiveCategory o ℓ e) where
  open ExtensiveDistributiveCategory D renaming (U to C; id to idC)
  open Cocartesian (Extensive.cocartesian extensive)

Definition 12: Uniform-Iteration Algebras

  record Uniform-Iteration-Algebra-on (A : Obj) : Set (o ⊔ ℓ ⊔ e) where
    -- iteration operator
    field
      _# : ∀ {X} → (X ⇒ A + X) → (X ⇒ A)

    -- _# properties
    field
      #-Fixpoint : ∀ {X} {f : X ⇒ A + X }
        → f # ≈ [ idC , f # ] ∘ f
      #-Uniformity : ∀ {X Y} {f : X ⇒ A + X} {g : Y ⇒ A + Y} {h : X ⇒ Y} 
        → (idC +₁ h) ∘ f ≈ g ∘ h
        → f # ≈ g # ∘ h
      #-resp-≈ : ∀ {X} {f g : X ⇒ A + X} → f ≈ g → (f #) ≈ (g #)

  record Uniform-Iteration-Algebra : Set (o ⊔ ℓ ⊔ e) where
    field
      A : Obj
      algebra : Uniform-Iteration-Algebra-on A
    open Uniform-Iteration-Algebra-on algebra public